Page 181 - Intro to Tensor Calculus

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EXAMPLE 2.1-4. (Maxwell’s equations) In the study of electrodynamics there arises the following
vectors and scalars:
~
~
E =Electric force vector, [E]= Newton/coulomb
~
~
B =Magnetic force vector, [B]= Weber/m 2
~
~
D =Displacement vector, [D] = coulomb/m 2
~
~
H =Auxilary magnetic force vector, [H] = ampere/m
~
~
J =Free current density, [J] = ampere/m 2
% =free charge density, [%] = coulomb/m 3
The above quantities arise in the representation of the following laws:
Faraday’s Law This law states the line integral of the electromagnetic force around a loop is proportional
to the rate of flux of magnetic induction through the loop. This gives rise to the first electromagnetic field
equation:
~
∂B ∂B i
~ ijk
∇× E = − or E k,j = − . (2.1.15)
∂t ∂t
Ampere’s Law This law states the line integral of the magnetic force vector around a closed loop is
proportional to the sum of the current through the loop and the rate of flux of the displacement vector
through the loop. This produces the second electromagnetic field equation:
~
∂D ijk i ∂D i
~
~
∇× H = J + or H k,j = J + . (2.1.16)
∂t ∂t
Gauss’s Law for Electricity This law states that the flux of the electric force vector through a closed
surface is proportional to the total charge enclosed by the surface. This results in the third electromagnetic
field equation:
1 ∂ √ i
~
∇· D = % or √ gD = %. (2.1.17)
g ∂x i
Gauss’s Law for Magnetism This law states the magnetic flux through any closed volume is zero. This
produces the fourth electromagnetic field equation:
1 ∂ √ i
~
∇· B =0 or √ gB =0. (2.1.18)
g ∂x i
The four electromagnetic field equations are referred to as Maxwell’s equations. These equations arise
in the study of electrodynamics and can be represented in other forms. These other forms will depend upon
such things as the material assumptions and units of measurements used. Note that the tensor equations
(2.1.15) through (2.1.18) are representations of Maxwell’s equations in a form which is independent of the
coordinate system chosen.
In applications, the tensor quantities must be expressed in terms of their physical components. In a
general orthogonal curvilinear coordinate system we will have

2
2
2
g 11 = h , g 22 = h , g 33 = h , and g ij =0 for i 6= j.
1 2 3
√
This produces the result g = h 1 h 2 h 3 . Further, if we represent the physical components of
D i ,B i ,E i ,H i by D(i),B(i),E(i), and H(i)